Fractal structures and scaling laws in the universe. Statistical mechanics of the self-gravitating gas

Abstract

Fractal structures are observed in the universe in two very different ways. Firstly, in the gas forming the cold interstellar medium in scales from 10⁻⁴pc till 100pc. Secondly, the galaxy distribution has been observed to be fractal in scales up to hundreds of Mpc. We give here a short review of the statistical mechanical (and field theoretical) approach developed by us for the cold interstellar medium (ISM) and large structure of the universe. We consider a non-relativistic self-gravitating gas in thermal equilibrium at temperature T inside a volume V. The statistical mechanics of such system has special features and, as is known, the thermodynamical limit does not exist in its customary form. Moreover, the treatments through microcanonical, canonical and grand canonical ensembles yield different results. We present here for the first time the equation of state for the self-gravitating gas in the canonical ensemble. We find that it has the form p = [NT/V]f(η), where p is the pressure, N is the number of particles and $\text{η ≡}\text{Gm}{}^2\text{N}\text{V}{}^{\mathrm{1/3}}\text{T}$. The N → ∞ and V → ∞ limit exists keeping η fixed. We compute the function f(η) using Monte Carlo simulations and for small η, analytically. We compute the thermodynamic quantities of the system as free energy, entropy, chemical potential, specific heat, compressibility and speed of sound. We reproduce the well-known gravitational phase transition associated to the Jeans' instability. Namely, a gaseous phase for η < η_c and a condensed phase for η > η_c. Moreover, we derive the precise behaviour of the physical quantities near the transition. In particular, the pressure vanishes as p ∼ (η_c − η)^B with B ∼ 0.2 and η_c ∼ 1.6 and the energy fluctuations diverge as ∼ (η_c − η)^B−1. The speed of sound decreases monotonically with η and approaches the value √(T/6) at the transition.